Found problems: 85335
TNO 2023 Senior, 5
Find all triples of integers \( (x, y, z) \) such that
\[
x - yz = 11
\]
\[
xz + y = 13
\]
2002 AMC 12/AHSME, 19
The graph of the function $ f$ is shown below. How many solutions does the equation $ f(f(x)) \equal{} 6$ have?
[asy]size(220);
defaultpen(fontsize(10pt)+linewidth(.8pt));
dotfactor=4;
pair P1=(-7,-4), P2=(-2,6), P3=(0,0), P4=(1,6), P5=(5,-6);
real[] xticks={-7,-6,-5,-4,-3,-2,-1,1,2,3,4,5,6};
real[] yticks={-6,-5,-4,-3,-2,-1,1,2,3,4,5,6};
draw(P1--P2--P3--P4--P5);
dot("(-7, -4)",P1);
dot("(-2, 6)",P2,LeftSide);
dot("(1, 6)",P4);
dot("(5, -6)",P5);
xaxis("$x$",-7.5,7,Ticks(xticks),EndArrow(6));
yaxis("$y$",-6.5,7,Ticks(yticks),EndArrow(6));[/asy]$ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 7$
2008 Silk Road, 1
Suppose $ a,c,d \in N$ and $ d|a^2b\plus{}c$ and $ d\geq a\plus{}c$
Prove that $ d\geq a\plus{}\sqrt[2b] {a}$
2012 Purple Comet Problems, 1
Last month a pet store sold three times as many cats as dogs. If the store had sold the same number of cats but eight more dogs, it would have sold twice as many cats as dogs. How many cats did the pet store sell last month?
2007 Junior Macedonian Mathematical Olympiad, 2
Let $ABCD$ be a parallelogram and let $E$ be a point on the side $AD$, such that $\frac{AE}{ED} = m$. Let $F$ be a point on $CE$, such that $BF \perp CE$, and the point $G$ is symmetrical to $F$ with respect to $AB$. If point $A$ is the circumcenter of triangle $BFG$, find the value of $m$.
1990 AMC 8, 15
The area of this figure is $ 100\text{ cm}^{2} $. Its perimeter is
[asy]
draw((0,2)--(2,2)--(2,1)--(3,1)--(3,0)--(1,0)--(1,1)--(0,1)--cycle,linewidth(1));
draw((1,2)--(1,1)--(2,1)--(2,0),dashed);[/asy]
$ \text{(A)}\ \text{20 cm}\qquad\text{(B)}\ \text{25 cm}\qquad\text{(C)}\ \text{30 cm}\qquad\text{(D)}\ \text{40 cm}\qquad\text{(E)}\ \text{50 cm} $
2016 PUMaC Number Theory A, 3
For odd positive integers $n$, define $f(n)$ to be the smallest odd integer greater than $n$ that is not relatively prime to $n$. Compute the smallest $n$ such that $f(f(n))$ is not divisible by $3$.
2024 New Zealand MO, 6
Let $\omega$ be the incircle of scalene triangle $ABC$. Let $\omega$ be tangent to $AB$ and $AC$ at points $X$ and $Y$. Construct points $X^\prime$ and $Y^\prime$ on line segments $AB$ and $AC$ respectively such that $AX^\prime=XB$ and $AY^\prime=YC$. Let line $CX^\prime$ intersects $\omega$ at points $P,Q$ such that $P$ is closer to $C$ than $Q$. Also let $R^\prime$ be the intersection of lines $CX^\prime$ and $BY^\prime$. Prove that $CP=RX^\prime$.
2022 CCA Math Bonanza, L1.4
Jongol and Gongol are writing calculus questions and grading tests. They want to write 90 calculus problems and they have 120 tests to grade. Jongol can write 3 questions per minute or grade 4 tests per minute. Gongol can write 1 question per minute or grade 2 tests per minute. Evaluate the shortest possible time, in minutes, for them to complete the two tasks.
[i]2022 CCA Math Bonanza Lightning Round 1.4[/i]
2003 Vietnam National Olympiad, 2
Define $p(x) = 4x^{3}-2x^{2}-15x+9, q(x) = 12x^{3}+6x^{2}-7x+1$. Show that each polynomial has just three distinct real roots. Let $A$ be the largest root of $p(x)$ and $B$ the largest root of $q(x)$. Show that $A^{2}+3 B^{2}= 4$.
VI Soros Olympiad 1999 - 2000 (Russia), 9.10
The schoolboy wrote a homework essay on the topic “How I spent my summer.” Two of his comrades from a neighboring school decided not to bother themselves with work and rewrote his essay. But while rewriting they made several mistakes - each their own. Before submitting their work, both students gave their essays to four other friends to rewrite (each gave them to two acquaintances). These four schoolchildren do the same, and so on. With each rewrite, all previous mistakes are saved and, possibly, new ones are made. It is known that on some day each new essay contained at least $10$ errors. Prove that there was a day when at least $11$ new mistakes were made in total.
2019 Iran MO (3rd Round), 2
Call a polynomial $P(x)=a_nx^n+a_{n-1}x^{n-1}+\dots a_1x+a_0$ with integer coefficients primitive if and only if $\gcd(a_n,a_{n-1},\dots a_1,a_0) =1$.
a)Let $P(x)$ be a primitive polynomial with degree less than $1398$ and $S$ be a set of primes greater than $1398$.Prove that there is a positive integer $n$ so that $P(n)$ is not divisible by any prime in $S$.
b)Prove that there exist a primitive polynomial $P(x)$ with degree less than $1398$ so that for any set $S$ of primes less than $1398$ the polynomial $P(x)$ is always divisible by product of elements of $S$.
2003 Junior Balkan Team Selection Tests - Romania, 1
Let $a, b, c$ be positive real numbers with $abc = 1$. Prove that $1 + \frac{3}{a+b+c}\ge \frac{6}{ab+bc+ca}$
2008 China Team Selection Test, 6
Find the maximal constant $ M$, such that for arbitrary integer $ n\geq 3,$ there exist two sequences of positive real number $ a_{1},a_{2},\cdots,a_{n},$ and $ b_{1},b_{2},\cdots,b_{n},$ satisfying
(1):$ \sum_{k \equal{} 1}^{n}b_{k} \equal{} 1,2b_{k}\geq b_{k \minus{} 1} \plus{} b_{k \plus{} 1},k \equal{} 2,3,\cdots,n \minus{} 1;$
(2):$ a_{k}^2\leq 1 \plus{} \sum_{i \equal{} 1}^{k}a_{i}b_{i},k \equal{} 1,2,3,\cdots,n, a_{n}\equiv M$.
2013 Kosovo National Mathematical Olympiad, 2
Math teacher wrote in a table a polynomial $P(x)$ with integer coefficients and he said:
"Today my daughter have a birthday.If in polynomial $P(x)$ we have $x=a$ where $a$ is the age of my daughter we have $P(a)=a$ and $P(0)=p$ where $p$ is a prime number such that $p>a$."
How old is the daughter of math teacher?
2023 IMAR Test, P1
Let $ABC$ be an acute triangle, and let $D,E,F$ be the feet of its altitudes from $A,B,C$ respectively. The lines $AB{}$ and $DE$ cross at $K{}$ and the lines $AC$ and $DF$ cross at $L{}.$ Let $M$ be the midpoint of the side $BC$ and let the line $AM$ cross the circle $(ABC)$ again at $N{}.$ The parallel through $M{}$ to $EF$ crosses the line $KL$ at $P{}.$ Prove that the triangle $MNP$ is isosceles.
2018 MOAA, 2
If $x > 0$ and $x^2 +\frac{1}{x^2}= 14$, find $x^5 +\frac{1}{x^5}$.
1985 Traian Lălescu, 1.3
Let $ a,b,c $ denote the lengths of a right triangle ($ a $ being the hypothenuse) that satisfy the equality $ a=2\sqrt{bc} . $
Find the angles of this triangle.
2023 Balkan MO Shortlist, G6
Let $ABC$ be an acute triangle ($AB < BC < AC$) with circumcircle $\Gamma$. Assume there exists $X \in AC$ satisfying $AB=BX$ and $AX=BC$. Points $D, E \in \Gamma$ are taken such that $\angle ADB<90^{\circ}$, $DA=DB$ and $BC=CE$. Let $P$ be the intersection point of $AE$ with the tangent line to $\Gamma$ at $B$, and let $Q$ be the intersection point of $AB$ with tangent line to $\Gamma$ at $C$. Show that the projection of $D$ onto $PQ$ lies on the circumcircle of $\triangle PAB$.
2018 Miklós Schweitzer, 5
For every positive integer $n$, define
$$f(n)=\sum_{p\mid n}{p^{k_p}},$$where the sum is taken over all positive prime divisors $p$ of $n$, and $k_p$ is the unique integer satisfying $$p^{k_p}\leqslant n<p^{k_p+1}.$$Find$$\limsup_{n\to \infty} \frac{f(n)\log \log n}{n\log n} .$$
2019-IMOC, C1
Given a natural number $n$, if the tuple $(x_1,x_2,\ldots,x_k)$ satisfies
$$2\mid x_1,x_2,\ldots,x_k$$
$$x_1+x_2+\ldots+x_k=n$$
then we say that it's an [i]even partition[/i]. We define [i]odd partition[/i] in a similar way. Determine all $n$ such that the number of even partitions is equal to the number of odd partitions.
1995 China Team Selection Test, 1
Find the smallest prime number $p$ that cannot be represented in the form $|3^{a} - 2^{b}|$, where $a$ and $b$ are non-negative integers.
1994 IberoAmerican, 2
Let $n$ and $r$ two positive integers. It is wanted to make $r$ subsets $A_1,\ A_2,\dots,A_r$ from the set $\{0,1,\cdots,n-1\}$ such that all those subsets contain exactly $k$ elements and such that, for all integer $x$ with $0\leq{x}\leq{n-1}$ there exist $x_1\in{}A_1,\ x_2\in{}A_2 \dots,x_r\in{}A_r$ (an element of each set) with $x=x_1+x_2+\cdots+x_r$.
Find the minimum value of $k$ in terms of $n$ and $r$.
2005 India IMO Training Camp, 3
Consider a matrix of size $n\times n$ whose entries are real numbers of absolute value not exceeding $1$. The sum of all entries of the matrix is $0$. Let $n$ be an even positive integer. Determine the least number $C$ such that every such matrix necessarily has a row or a column with the sum of its entries not exceeding $C$ in absolute value.
[i]Proposed by Marcin Kuczma, Poland[/i]
2005 Georgia Team Selection Test, 8
In a convex quadrilateral $ ABCD$ the points $ P$ and $ Q$ are chosen on the sides $ BC$ and $ CD$ respectively so that $ \angle{BAP}\equal{}\angle{DAQ}$. Prove that the line, passing through the orthocenters of triangles $ ABP$ and $ ADQ$, is perpendicular to $ AC$ if and only if the triangles $ ABP$ and $ ADQ$ have the same areas.